Investments and Valuation: cram pack

Formula sheets, the key method steps for every topic and 8 practice problems, with full solutions at the back.

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1Formulas and methods

Time value of money

Future value: FV = PV·(1 + i)ⁿ
Present value: PV = FV/(1 + i)ⁿ
Several periods per year: i = r/m, N = m·n
Annuity, present value: PV = PMT·[1 − (1 + i)⁻ⁿ]/i
Annuity, future value: FV = PMT·[(1 + i)ⁿ − 1]/i

Method

Draw a timeline with every cash flow and its timing.
Match the rate to the period: a monthly rate for monthly payments.
Use FV = PV·(1 + i)ⁿ for single amounts and the annuity formulas for equal payments.
Check the sign and the order of magnitude of the answer.

Bond pricing & yield

Bond price: P = C·(1 − (1 + y)⁻ⁿ)/y + F/(1 + y)ⁿ
Semi-annual: C/2, y/2 and 2n periods
Current yield: C/P
Price and yield: yield up → price down

Method

Find the coupon per period and the yield per period.
Price = present value of the coupons (an annuity) + present value of the face value.
For semi-annual coupons use half the coupon, half the yield and twice the periods.
Check: if the yield is above the coupon rate, the price is below face value.

CAPM & WACC

CAPM: r_e = r_f + β·(r_m − r_f)
WACC: E/V·r_e + D/V·r_d·(1 − T)
Firm value: V = E + D

NPV & IRR

Net present value: NPV = −C₀ + Σ CF_t/(1 + r)^t
Equal cash flows: NPV = −C₀ + CF·[1 − (1 + r)⁻ⁿ]/r
Internal rate of return: the r at which NPV = 0
Decision: accept when NPV > 0 (or IRR > required return)

Method

List all cash flows with their year; the investment at year 0 is negative.
Discount each cash flow: CF_t/(1 + r)^t.
Add them up to get the NPV. Positive means the project earns more than r.
For the IRR, find the rate that makes NPV zero by trial or interpolation.

2Practice problems

1.Time value of moneyMedium

You need €10 000 in 5 years. An account pays 4.8% per year, compounded monthly. How much must you deposit today? Find the present value PV.

FV = €10 0004.8% / 12n = 5

PV = €

2.Time value of moneyHard

You save €200 at the end of every month for 10 years. The account pays 4.8% per year, compounded monthly. How much have you saved at the end? Find FV.

A = €2004.8% / 12N = 120

FV = €

3.Bond pricing & yieldMedium

A bond with a face value of €1 000 pays an annual coupon of 4.5% and matures in 4 years. Its yield to maturity changes from 5.5% to 7.5%. By how much ΔP does the bond price change?

F = €1 000coupon = 4.5%n = 4y₁ = 5.5%y₂ = 7.5%

ΔP = €

4.Bond pricing & yieldHard

A bond with a face value of €1 000 pays an annual coupon of 3% and matures in 3 years. The yield to maturity is 8%. Find the Macaulay duration D in years.

F = €1 000coupon = 3%n = 3YTM = 8%

D = years

5.CAPM & WACCMedium

A company has equity worth €4 000 000 and debt worth €2 000 000. Its WACC is 10.73%, its pre-tax cost of debt is 6% and the tax rate is 30%. Find the cost of equity r_e.

E = €4 000 000D = €2 000 000WACC = 10.73%r_d = 6%T = 30%

r_e = %

6.CAPM & WACCHard

A company's business has an unlevered (asset) beta of 1.2. It plans a debt-to-equity ratio of 1. The tax rate is 20%, the risk-free rate is 1.5% and the market risk premium is 7%. Find the cost of equity r_e at the new capital structure.

β_U = 1.2D/E = 1T = 20%r_f = 1.5%r_m − r_f = 7%

r_e = %

7.NPV & IRRMedium

An investment of €45 000 gives these cash flows at the end of years 1 to 4: €16 000, €18 000, €20 000, €11 000. The cost of capital is 8%. Find the net present value NPV.

r = 8%CF₀ = −45 000CF₁ = 16 000CF₂ = 18 000CF₃ = 20 000CF₄ = 11 000

NPV = €

8.NPV & IRRHard

An investment of €40 000 returns €8 500 at the end of each year for 6 years. Find its internal rate of return IRR in percent.

C₀ = €40 000CF = €8 500n = 6

IRR = %

3Solutions

1.Time value of money

Answer: PV = 7870 €

Time value of money: money today is worth more than the same amount later, because it can earn interest. With interest rate i per period, an amount grows as FV = PV·(1 + i)ⁿ. A series of equal payments (an annuity) is valued by adding up each payment, which gives the annuity formulas.

Rate and number of periods

i = 0.048/12 = 0.004

N = 12·5 = 60

Present value

PV = FV/(1 + i)^N = 10 000/1.271 = €7 870.05

Answer

PV = €7 870.05

2.Time value of money

Answer: FV = 30726 €

Monthly rate and number of deposits

i = 0.048/12 = 0.004

N = 12·10 = 120

Future value of the deposits

FV = A·((1 + i)^N − 1)/i = 200·(1.615 − 1)/0.004 = €30 726.39

Answer

FV = €30 726.39

You paid in €24 000 yourself; the rest is interest.

3.Bond pricing & yield

Answer: ΔP = −65.43 €

Bond pricing: a bond’s price is the present value of its cash flows, discounted at the yield to maturity (YTM): the coupons form an annuity and the face value is paid back at maturity. P = C·(1 − (1 + y)⁻ⁿ)/y + F/(1 + y)ⁿ. When the yield is above the coupon rate the bond trades below face value, and the other way round.

Price at the old yield

C = €45.00, i = 0.055, N = 4

€45.00·(1 − 0.8072)/0.055 + €1 000·0.8072 = €157.73 + €807.22 = €964.95

Price at the new yield

C = €45.00, i = 0.075, N = 4

€45.00·(1 − 0.7488)/0.075 + €1 000·0.7488 = €150.72 + €748.80 = €899.52

Change

ΔP = 899.52 − 964.95 = −€65.43

Answer

ΔP = −€65.43

4.Bond pricing & yield

Answer: D = 2.907 years

Duration: the Macaulay duration is the weighted average time at which a bond pays its cash flows, with the present values of the cash flows as weights: D = Σ t·PVₜ/P. The modified duration D/(1 + y) estimates the percentage price change for a small change in yield: %ΔP ≈ −D_mod·Δy.

Present value of each cash flow

t = 1: 30/1.08 = 27.78, t·PV = 27.78

t = 2: 30/1.166 = 25.72, t·PV = 51.44

t = 3: 1030/1.26 = 817.65, t·PV = 2 452.94

Duration

P = 871.15

D = 2 532.16/871.15 = 2.907

Answer

D = 2.907 years

5.CAPM & WACC

Answer: r_e = 14 %

Cost of capital: CAPM gives the return shareholders require: r_e = r_f + β·(r_m − r_f). The weighted average cost of capital mixes equity and debt by their market values: WACC = E/V·r_e + D/V·r_d·(1 − T), with V = E + D. Interest is tax-deductible, hence the (1 − T).

Weights

V = €6 000 000, E/V = 0.6667, D/V = 0.3333

Debt part of the WACC

0.3333·6%·(1 − 0.3) = 1.4%

Cost of equity

r_e = (10.73% − 1.4%)/0.6667 = 14%

Answer

r_e = 14%

6.CAPM & WACC

Answer: r_e = 16.62 %

Levered beta (Hamada): debt makes equity riskier. β_L = β_U·(1 + (1 − T)·D/E), where β_U is the beta of the business without debt. To move a beta to another capital structure, unlever it first and then relever it at the new D/E.

Levered beta

β_L = 1.2·(1 + 0.8·1) = 2.16

Cost of equity (CAPM)

r_e = 1.5% + 2.16·7% = 16.62%

Answer

r_e = 16.62%

7.NPV & IRR

Answer: NPV = 9209 €

Net present value: discount every cash flow back to today with the required rate of return r, then add them up: NPV = Σ CFₜ/(1 + r)ᵗ. A project with NPV > 0 earns more than r and adds value. The internal rate of return (IRR) is the rate at which NPV = 0.

Present value of each cash flow

CF₀ = −45 000

CF₁/(1 + 0.08)¹ = 16 000/1.08 = 14 814.81

CF₂/(1 + 0.08)² = 18 000/1.166 = 15 432.10

CF₃/(1 + 0.08)³ = 20 000/1.26 = 15 876.64

CF₄/(1 + 0.08)⁴ = 11 000/1.36 = 8 085.33

Add them up

NPV = €9 208.89

Answer

NPV = €9 208.89

NPV is positive, so the project earns more than the required return.

8.NPV & IRR

Answer: IRR = 7.416 %

Required annuity factor

(1 − (1 + i)⁻⁶)/i = 40 000/8 500 = 4.706

Solve for the rate

The annuity factor falls as the rate rises; bracket the rate and narrow down.

i = 7%: 4.767

i = 8%: 4.623

IRR = 7.416%

Answer

IRR = 7.416%

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